A very long cylindrical capacitor consists of two thin hollow conducting cylinders with the same axis of symmetry. The inner cylinder has a radius a, the outer one has a radius b. You may ignore end effects.
(a) Fig. 1
(b) Fig. 2
(a) What is the capacitance per unit length? Express your answer in terms of a,band epsilon_0
incorrect
(b) Now consider the limit where b is very close to a . Express b as a+δ ; where δ /a<<1 . What is the capacitance per unit length in that limit? hint: in that limit you can use the following approximation:
ln(1+x)≈ x
Express your answer in terms of a, and δ
Please someone answer.
To find the capacitance per unit length of the cylindrical capacitor in part (a), we can use the formula for the capacitance of a cylindrical capacitor:
C = (2πε₀) / ln(b / a)
Where:
C is the capacitance per unit length,
ε₀ is the permittivity of free space,
a is the radius of the inner cylinder,
b is the radius of the outer cylinder.
Now, let's move on to part (b), where b is very close to a. In this case, we can make use of the approximation ln(1 + x) ≈ x, which holds for small values of x.
Since b is very close to a, we can express b as a + δ, where δ / a << 1. Let's substitute this value into the original formula:
C = (2πε₀) / ln((a + δ) / a)
Now, using the approximation mentioned earlier:
C ≈ (2πε₀) / ((a + δ) / a)
C ≈ (2πε₀a) / (a + δ)
So, in the limit where b is very close to a, the capacitance per unit length can be approximated as (2πε₀a) / (a + δ).